arXiv subjects
Publications and source records attributed to Michael Leinert.
The spaces $BV(σ)$ and $AC(σ)$ were introduced as part of a program to find a general theory which covers both well-bounded operators and trigonometrically well-bounded operators acting on a Banach space. Since their initial appearance it has become clear that the definitions could be simplified somewhat. In this paper we give a self-contained exposition of the main properties of these spaces using this simplified approach.
For a nonempty compact subset $σ$ in the plane, the space $AC(σ)$ is the closure of the space of complex polynomials in two real variables under a particular variation norm. In the classical setting, $AC[0,1]$ contains several other useful dense subsets, such as continuous piecewise linear functions, $C^1$ functions and Lipschitz functions. In this paper we examine analogues of these results in this more general setting.
If $G$ is a locally compact group, $CD(G)$ the algebra of convolution dominated operators on $L^2(G)$ then an important question is: Is $\mathbb{C}1+CD(G)$ (respectively $CD(G)$ if $G$ is discrete) inverse-closed in the bounded operators on $L^2(G)$? In this note we answer this question in the affirmative provided $G$ is such that one of the following properties is fulfilled (1) There is a discrete, rigidly symmetric, and amenable subgroup $H\subset G$ and a (measurable) relatively compact neighbourhood of the identity $U$ invariant under conjugation by elements of $H$ such that $\{hU\;:\;h\in H\}$ is a partition of $G$. (2) The commutator subgroup of $G$ is relatively compact. (If $G$ is connected this just means that $G$ is an IN group.) All known examples where $CD(G)$ is inverse-closed in $B(L^2(G))$ are covered by this.
For a locally compact group $G$ we consider the algebra $CD(G)$ of convolution dominated operators on $L^{2}(G)$: An operator $A:L^2(G)\to L^2(G)$ is called convolution dominated if there exists $a\in L^1(G)$ such that for all $f \in L^2(G)$ $ |Af(x)| \leq a * |f| (x)$, for almost all $x \in G$. In the case of discrete groups those operators can be dealt with quite sufficiently if the group in question is rigidly symmetric. For non-discrete groups we investigate the subalgebra of regular convolution dominated operators $CD_{reg}(G)$. For amenable $G$ which is rigidly symmetric as a discrete group, we show that any element of $CD_{reg}(G)$ is invertible in $CD_{reg}(G)$ if it is invertible as a bounded operator on $L^2(G)$. We give an example of a symmetric group $E$ for which the convolution dominated operators are not inverse-closed in the bounded operators on $L^2(E)$.
The following theorem on the circle group $\mathbb{T}$ is due to Norbert Wiener: If $f\in L^{1}\left( \mathbb{T}\right) $ has non-negative Fourier coefficients and is square integrable on a neighbourhood of the identity, then $f\in L^{2}\left( \mathbb{T}\right) $. This result has been extended to even exponents including $p=\infty$, but shown to fail for all other $p\in\left( 1,\infty\right] .$ All of this was extended further (appropriately formulated) well beyond locally compact abelian groups. In this paper we prove Wiener's theorem for even exponents for a large class of commutative hypergroups. In addition, we present examples of commutative hypergroups for which, in sharp contrast to the group case, Wiener's theorem holds for all exponents $p\in\left[ 1,\infty\right] $. For these hypergroups and the Bessel-Kingman hypergroup with parameter $\frac{1}{2}$ we characterise those locally integrable functions that are of positive type and square-integrable near the identity in terms of amalgam spaces.
Analogues of the classical Banach-Stone theorem for spaces of continuous functions are studied in the context of the spaces of absolutely continuous functions introduced by Ashton and Doust. We show that if $AC(σ_1)$ is algebra isomorphic to $AC(σ_2)$ then $σ_1$ is homeomorphic to $σ_2$. The converse however is false. In a positive direction we show that the converse implication does hold if the sets $σ_1$ and $σ_2$ are confined to a restricted collection of compact sets, such as the set of all simple polygons.
It is shown that the dual $\hat{A}$ of a separable $C^{\ast}$-algebra $A$ is discrete if and only if its Banach space dual has the weak$^{\ast}$-fixed point property. We prove further that these properties are equivalent to the uniform weak$^{\ast}$ Kadec-Klee property of $A^{\ast}$ and to the coincidence of the weak$^{\ast}$ topology with the norm topology on the pure states of $A$.
We give a systematic construction of inverse-closed (Banach) subalgebras in general higher-dimensional non-commutative tori
We study infinite matrices $A$ indexed by a discrete group $G$ that are dominated by a convolution operator in the sense that $|(Ac)(x)| \leq (a \ast |c|)(x)$ for $x\in G$ and some $a\in \ell ^1(G)$. This class of "convolution-dominated" matrices forms a Banach-*-algebra contained in the algebra of bounded operators on $\ell ^2(G)$. Our main result shows that the inverse of a convolution-dominated matrix is again convolution-dominated, provided that $G$ is amenable and rigidly symmetric. For abelian groups this result goes back to Gohberg, Baskakov, and others, for non-abelian groups completely different techniques are required, such as generalized $L^1$-algebras and the symmetry of group algebras.
We introduce the notion of a conditionally free product and conditionally free convolution. We describe this convolution both from a combinatorial point of view, by showing its connection with the lattice of non-crossing partitions, and from an analytic point of view, by presenting the basic formula for its $R$-transform. We calculate explicitly the distributions of the conditionally free Gaussian and conditionally free Poisson distribution.